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October 2019 Inference for the mode of a log-concave density
Charles R. Doss, Jon A. Wellner
Ann. Statist. 47(5): 2950-2976 (October 2019). DOI: 10.1214/18-AOS1770


We study a likelihood ratio test for the location of the mode of a log-concave density. Our test is based on comparison of the log-likelihoods corresponding to the unconstrained maximum likelihood estimator of a log-concave density and the constrained maximum likelihood estimator where the constraint is that the mode of the density is fixed, say at $m$. The constrained estimation problem is studied in detail in Doss and Wellner (2018). Here, the results of that paper are used to show that, under the null hypothesis (and strict curvature of $-\log f$ at the mode), the likelihood ratio statistic is asymptotically pivotal: that is, it converges in distribution to a limiting distribution which is free of nuisance parameters, thus playing the role of the $\chi_{1}^{2}$ distribution in classical parametric statistical problems. By inverting this family of tests, we obtain new (likelihood ratio based) confidence intervals for the mode of a log-concave density $f$. These new intervals do not depend on any smoothing parameters. We study the new confidence intervals via Monte Carlo methods and illustrate them with two real data sets. The new intervals seem to have several advantages over existing procedures. Software implementing the test and confidence intervals is available in the R package \verb+logcondens.mode+.


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Charles R. Doss. Jon A. Wellner. "Inference for the mode of a log-concave density." Ann. Statist. 47 (5) 2950 - 2976, October 2019.


Received: 1 December 2016; Revised: 1 October 2018; Published: October 2019
First available in Project Euclid: 3 August 2019

zbMATH: 07114934
MathSciNet: MR3988778
Digital Object Identifier: 10.1214/18-AOS1770

Primary: 62G07
Secondary: 62G10, 62G15, 62G20

Rights: Copyright © 2019 Institute of Mathematical Statistics


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Vol.47 • No. 5 • October 2019
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